1.1.1 Goals of this lecture

• Interactions in GLiM
  • GLiMs have nonlinear (conditional) effects
  • Interactions are also conditional effects
  • How do those work together?
• Mediation with GLiM
  • Continuous outcomes: Indirect effects are calculated as the product of two regression coefficients from linear regression
  • GLiMs are nonlinear, so what do we use?

2.1.1 Multiple predictors with no interaction

\[\hat{Y} = b_0 + b_1 X_1 + b_2 X_2\]

• \(b_0\) is the intercept
  • \(\hat{Y}\) when both \(X_1\) and \(X_2\) are equal to 0
• \(b_1\) is the (partial) effect of \(X_1\)
  • The effect of \(X_1\) on \(\hat{Y}\), holding all other predictors constant
• \(b_2\) is the (partial) effect of \(X_2\)
  • The effect of \(X_2\) on \(\hat{Y}\), holding all other predictors constant

2.1.2 Figure: 1 continuous and 1 binary, no interaction

2.1.3 Interaction as product term

\[\hat{Y} = b_0 + b_1 X_1 + b_2 X_2 + b_3 X_1 X_2\]

• \(b_0\) is the intercept
  • \(\hat{Y}\) when both \(X_1\) and \(X_2\) are equal to 0
• \(b_3\) is the interaction term
  • How the effect of \(X_1\) on \(\hat{Y}\) varies as a function of \(X_2\)
  • How the effect of \(X_2\) on \(\hat{Y}\) varies as a function of \(X_1\)

2.1.4 Simple slopes

\[\hat{Y} = (\color{red}{b_0 + b_2 X_2}) + (\color{blue}{b_1 + b_3 X_2}) X_1\]

• Select specific values of \(X_2\) and simplify
  • Determined by variable (binary), values of interest (mean, median, cut-offs), \(\pm\) 1 standard deviation (Aiken and West, 1991)
  • Each value of \(X_2\) results in a specific intercept and slope
    • Intercept: \(\color{red}{b_0 + b_2 X_2}\)
    • Slope: \(\color{blue}{b_1 + b_3 X_2}\)

2.1.5 Figure: 1 continuous, 1 binary w/ interaction

2.1.6 Conditional effects

• Simple slopes are conditional effects
  • The effect of \(X_1\), conditional on the fact that \(X_2\) takes on a certain value
  • Not a single value
  • Varies across values of a variable (here, \(X_2\))

2.1.7 Marginal effect

• We can also talk about the marginal effect of the interaction
  • Single value that reflects the overall effect
  • For linear regression, this is \(b_3\)
    • For 1 continuous and 1 binary predictor, this is the difference in slopes between the groups
    • For 2 continuous predictors, this is the warp in the 3D regression plane away from flat

2.2.1 How are interactions different in GLiMs?

• Everything we know about interactions from linear regression still applies
  • But only for linear metric of the GLiM
    • Logit metric for logistic regression
    • ln(count) metric for Poisson regression
  • We generally use other metrics
    • Probability metric for logistic regression
    • Count metric for Poisson regression

2.2.2 How are interactions different in GLiMs?

• Nonlinear GLiM effects without interaction are already conditional
  • Interaction effects are “doubly conditional”
• Interaction depends on more than 1 coefficient
  • No single, marginal interaction effect
• Product term is neither necessary nor sufficient to demonstrate interaction

2.2.3 GLiM effects without interaction are already conditional

• Logistic regression w single predictor: Slope depends on the predictor
  • No single number for the slope (i.e., linear change in outcome)
  • It varies depending on the predictor
• Poisson regression w single predictor: Slope depends on the predictor
  • No single number for the slope (i.e., linear change in outcome)
  • It varies depending on the predictor
• Even without any interaction, effects are conditional on the predictor

2.2.4 Interaction effects are “doubly conditional”

• We can create simple slopes for a GLiM, similar to linear regression
  • Conditional effects
• But now they’re conditional on both predictors
  • The “moderator” variable (usually \(X_2\))
  • The “focal” or “X axis” variable (usually \(X_1\))

2.2.5 Interaction depends on more than 1 coefficient

• Remember the odds ratio (\(e^{b_1}\) for \(X_1\))
  • This is the change in \(\hat{Y}\) due to change in \(X_1\)
  • But it tells you nothing about where you start and end
  • OR = 2 could be odds of 2 versus 1, or odds of 10 versus 5
  • Have to look at \(b_0\) and \(b_1\) together
• Similarly, with interactions, you can’t look at the \(b_3\) coefficient alone
  • Several coefficients together tell you about the interaction

2.2.6 What is “the” interaction effect?

• tl;dr: It’s really super complicated and isn’t a single value

• The interaction is conditional on values of both \(X_1\) and \(X_2\)

  • No single value

• What to do?

  • Evaluate the interaction effect across different values of both predictors

2.2.7 Do I even need the product term?

• tl;dr: Maybe, but also maybe not

• The product term is neither necessary nor sufficient to determine whether there’s an interaction

  • See last slide: Interaction isn’t about just product term coefficient anyway

• Compare a model with a product term to a model without a product term

  • Use the better model (based on LR test)
  • Even the model without a product term will have a doubly conditional effect

2.3.1 Part 1. Determine the best model

• Compare model with product term to one without using LR test
  • It may be a model with a product term, but it may not
  • Regardless of which model is best, you can still have an interaction

2.3.2 Part 2. Equations for marginal effect are hard

• The marginal interaction effect is the second derivative of the regression equation with respect to both \(X_1\) and \(X_2\)
  • Exactly what that looks like depends on whether the predictors are continuous or categorical
• Don’t try to do the math here
  • Unless you are very comfortable with calculus
  • They are just to give you an idea of how it works

2.3.3 Why derivatives?

2.3.4 Equations for marginal effect

• Two continuous predictors: \[\beta_{3} \dot{g}^{-1}(d(x)^T \beta + (\beta_1 + \beta_3 X_2)(\beta_2 + \beta_3 X_1) \ddot{g}^{-1}(d(x)^T \beta) \]

• One continuous, one categorical predictor: \[(\beta_2 + \beta_3) \dot{g}^{-1}((\beta_2 + \beta_3) X_2 + \beta_0 + \beta_1) - \beta_2 \dot{g}^{-1}(\beta_0 + \beta_2 X_2)\]

• Two categorical predictors: \[\dot{g}^{-1}(\beta_0 + \beta_1 + \beta_2 + \beta_3) - \dot{g}^{-1}(\beta_0 + \beta_1) - \dot{g}^{-1} (\beta_0 + \beta_2) + \dot{g}^{-1}(\beta_0)\]

2.3.5 Equations for marginal effect are hard

• Notice that the interaction is a function of a bunch of things: \(\beta_1\), \(\beta_2\), \(\beta_3\), as well as other covariates in the model (\(\beta\))

• There are also first and second derivatives of the inverse link function

  • Depends on the model (i.e., logistic, Poisson, etc.)

2.3.6 Part 3. Conditional effects are (relatively) easy

• Conditional effects = simple slopes

• With the caveat that, for GLiMs, they are also conditional on the predictor

• As we’ve already seen for GLiMs

2.4.1 Example data

• Simulated data
  • case: Subject ID
  • sensation: Sensation seeking (1 to 7)
  • gender: 0 = female, 1 = male
  • y: Number of alcoholic beverages consumed on Saturday night

Coxe, S., West, S. G., & Aiken, L. S. (2009). The analysis of count data: A gentle introduction to Poisson regression and its alternatives. Journal of Personality Assessment, 91(2), 121-136.

2.4.2 Do we need a product term?

Call:
glm(formula = y ~ sensation4 + gender, family = poisson(link = "log"), 
    data = jpa)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-3.3696  -1.5739  -0.4401   0.8383   3.8439  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)  0.25455    0.07434   3.424 0.000617 ***
sensation4   0.26085    0.03882   6.719 1.83e-11 ***
gender       0.83947    0.06292  13.342  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 1186.76  on 399  degrees of freedom
Residual deviance:  959.46  on 397  degrees of freedom
AIC: 1888.8

Number of Fisher Scoring iterations: 5

Call:
glm(formula = y ~ sensation4 + gender + gender * sensation4, 
    family = poisson(link = "log"), data = jpa)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-3.4817  -1.7195  -0.5326   0.8161   3.7184  

Coefficients:
                  Estimate Std. Error z value Pr(>|z|)    
(Intercept)        0.46454    0.10909   4.258 2.06e-05 ***
sensation4         0.10318    0.07412   1.392   0.1639    
gender             0.55540    0.12950   4.289 1.80e-05 ***
sensation4:gender  0.21434    0.08694   2.465   0.0137 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 1186.76  on 399  degrees of freedom
Residual deviance:  953.47  on 396  degrees of freedom
AIC: 1884.8

Number of Fisher Scoring iterations: 5

2.4.3 Do we need a product term? Count metric

2.4.4 Do we need a product term? ln(count) metric

2.4.5 Do we need a product term? LR test

library(lmtest)
lrtest(jpa_m1, jpa_m2)

Likelihood ratio test

Model 1: y ~ sensation4 + gender
Model 2: y ~ sensation4 + gender + gender * sensation4
  #Df LogLik Df  Chisq Pr(>Chisq)  
1   3 -941.4                       
2   4 -938.4  1 5.9874    0.01441 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

2.4.6 Simple slopes: Conditional effects

\[ln(\hat{\mu}) = e^{(\color{red}{0.465 + 0.555 gender}) + (\color{blue}{0.103 + 0.214 gender}) sensation4}\]

• gender = 0
  • \(ln(\hat{\mu}) = e^{(\color{red}{0.465}) + (\color{blue}{0.103}) sensation4}\)
    • RR = \(e^{\color{blue}{0.103}} = 1.109\)
• gender = 1
  • \(ln(\hat{\mu}) = e^{(\color{red}{1.02}) + (\color{blue}{0.318}) sensation4}\)
    • RR = \(e^{\color{blue}{0.318}} = 1.374\)

2.4.7 Plot: Conditional effects

2.4.8 Compare predicted counts between groups

• When sensation4 = 0 (sensation = 4):
  • gender = 0: Predicted count = 1.59
  • gender = 1: Predicted count = 2.77
• When sensation4 = 1 (sensation = 5):
  • gender = 0: Predicted count = 1.76
  • gender = 1: Predicted count = 3.81
• When sensation4 = 2 (sensation = 6):
  • gender = 0: Predicted count = 1.96
    
  • gender = 1: Predicted count = 5.23

2.4.9 No constant effect across groups

2.4.10 modglm package

• McCabe, C. J., Halvorson, M. A., King, K. M., Cao, X., & Kim, D. S. (2020). Interpreting interaction effects in generalized linear models of nonlinear probabilities and counts. Multivariate Behavioral Research, 1-27. doi: https://doi.org/10.1080/00273171.2020.1868966

  • https://github.com/connorjmccabe/modglm

• I have not been able to get this to work for this dataset, but it does work with the simulated data they provide

3.1.1 Mediation model

3.1.2 Mediation equations

a path: \[\hat{M} = i_{MX} + aX\]

b and c’ paths: \[\hat{Y} = i_{YXM} + bM + c'X\]

c path: \[\hat{Y} = i_{YX} + cX\]

3.1.3 Mediated effect as product

3.1.4 Mediated effect as product

3.1.5 Mediated effect as product

• The mediated effect is the effect of X on Y via M
  • In SEM, such a path is described as the product of the regression coefficients that go into it
  • The \(a\) coefficient reflects the \(X \rightarrow M\) path
    • Slope for \(X\) predicting \(M\)
  • The \(b\) coefficient reflects the \(M \rightarrow Y\) path
    • Slope for \(M\) predicting \(Y\)
  • The mediated effect is \(a \times b\)

3.2.1 What is a slope?

3.2.2 What is a slope?

3.3.1 Nonlinear mediation

• Geldhof, G. J., Anthony, K. P., Selig, J. P., & Mendez-Luck, C. A. (2018). Accommodating binary and count variables in mediation: A case for conditional indirect effects. International Journal of Behavioral Development, 42(2), 300-308.

3.3.2 Mediated effect is still a product

• Still consider the mediated effect the product of two paths:
  • \(X\) to \(M\)
  • \(M\) to \(Y\)
• But what do we want from each of those path now?
  • Not just the “slope” (i.e., \(b_1\))
  • Depends on the specific model (i.e., linear, logistic, Poisson)

3.3.3 Slopes as derivatives

• Slopes are the first derivative of their respective equations

  • In linear regression, slopes simplify to \(a\) and \(b\)

  • In GLiMs, slopes are more complex

    • Use the appropriate derivative for your model

3.3.4 Derivatives for each model: \(X\) to \(M\)

• Table 1 from Geldhof et al. (2018)

• Note: \(i\) in the table refers to \(i_{XM}\): The intercept for \(X\) predicting \(M\)

3.3.5 Derivatives for each model: \(M\) to \(Y\)

• Table 2 from Geldhof et al. (2018)

• Note: \(i\) in the table refers to \(i_{YXM}\): The intercept for \(X\) and \(M\) predicting \(Y\)

3.4.1 Mediation example: JPA

3.4.2 \(X\) to \(M\): gender to sensation

• Linear regression

(Intercept)      gender 
   1.242024   -0.118866 

• \(a\) = \(-0.119\)

3.4.3 \(X\) to \(M\): gender to sensation

3.4.4 \(M\) to \(Y\): sensation to y (drinks)

• Poisson regression

(Intercept)  sensation4      gender 
  0.2545520   0.2608472   0.8394682 

• \(b e^{(i + bM + c'X)}\) = \(0.261 e^{(0.255 + 0.261 M + 0.839 X)}\)

3.4.5 \(M\) to \(Y\): sensation to y (drinks)

3.4.6 Mediated effect

• Product of two effects
  • \(a \times b \times e^{(i + bM + c'X)} = -0.119 \times 0.261 \times e^{(0.255 + 0.261 M + 0.839 X)}\)
  • Function of \(X\) (gender) and \(M\) (sensation4)
    • Conditional on \(X\) and \(M\)

3.4.7 Conditional indirect effect

• Select values of \(X\) based on the variable
  • 0 and 1
• Predict values of \(M\) based on the equation for \(X\) predicting \(M\)
  • See table
• Calculate mediated or indirect effect value based on \(X\) and \(M\)

  x        m         ind
1 0 1.242024 -0.05529634
2 1 1.123158 -0.12411010

3.4.8 Conditional indirect effect

• Mediated effect of gender to y (drinks) via sensation4
  • -0.055 for women (gender = 0)
  • -0.124 for men (gender = 1)
• Compare to incorrect, non-conditional approach
  • \(a \times b = -0.119 \times 0.261 = -0.031\)
  • Ignores conditional aspect
  • Ignores that the two paths come from different types of models

4.1.1 Summary of this week

• Interactions with GLiMs are hard
  • Use conditional effects (simple slopes)
  • Marginal effects are very complex
• Mediation with GLiMs is a little more difficult, but not too bad
  • Conditional indirect effect

4.1.2 Summary of this section

• GLiMs for binary, ordered and unordered categories, counts
  • Nonnormal outcomes
  • Nonlinear models with link functions
  • Different metrics
• Extensions of GLiMs
  • Interactions
  • Mediation

4.1.3 Next week

• Contingency tables
  • “Crosstabs” or frequency tables